Wavefront estimation, or equivalently wavefront reconstruction, from measured wavefront slope data is a classic problem in optical testing, active/adaptive optics, and media turbulence characterizations. It converts the wavefront slope data to wavefront optical path differences (OPDs) or wavefront phase estimates by multiplying the OPDs by 2π/λ. The OPDs shall be referred to as the wavefront values. The wavefront slope data is obtained from a slope wavefront sensor, and the task is to find a solution to the Neumann boundary problem of Poisson's equation.
Mathematical methods and algorithms for wavefront reconstruction in optical testing have been contributed by many authors. Approaches to wavefront reconstruction from slope data can be categorized as either zonal or modal estimation. In modal estimation, the wavefront is estimated by computing the coefficients of an aperture function with an orthogonal basis, whereas in zonal estimation, the wavefront is estimated by evaluating the wavefront values in local zones. In either case, a wavefront reconstruction is a least-squares estimate to the wavefront values, a numerical solution to the Neumann boundary problem. Only certain algorithms can handle general pupil shapes. These algorithms can be categorized into the Fourier-transform (FT)-based algorithms and the linear least-squares (LS)-based algorithms. By way of example, Gerchberg et al. pioneered the iterative Fast Fourier-Transform (FFT)-based phase retrievals from amplitude measurements in the aperture and the image planes in “A practical algorithm for the determination of phase from image and diffraction plane pictures”, as published in Optik, Volume 35, pages 237–246, 1972. Freischlad et al. disclosed a Discrete Fourier Transform (DFT)-based algorithm for zonal estimation from wavefront slope measurements for square-shaped pupils in “Wavefront Reconstruction From Noisy Slope or Difference Data Using the Discrete Fourier Transform”, published in Adaptive Optics, J. E. Ludman, ed., Proceedings of the SPIE, Volume 551, pages 74–80, 1985; and in “Modal Estimation of a Wavefront Difference Measurements Using the Discrete Fourier Transform”, published in the Journal of the Optical Society of America A, Volume 3, No. 11, pages 1852–1861, 1986.
Later, Freischlad extended this algorithm for general pupil shapes as described in “Wavefront Integration From Difference Data”, as published in Interferometry: Techniques and Analysis, G. M. Brown, O. Y. Kwon, M. Kujawinska, and G. T. Reid, Eds., Proceedings of the SPIE, Volume 1755, pages 212–218, 1992. In Freischlad's method, additional Least-Squares matrix equations needed to be set up to generate the missing slope data for extending the general shaped pupil to a square one. Roddier et al. disclosed a technique to extrapolate the wavefront outside of the pupil employing Gerchberg-type iterations and obtained an excellent FFT-based algorithm for irregular shaped pupils in “Interferogram Analysis Using Fourier Transform Techniques”, as published in Applied Optics, Volume 26, No. 9, pages 1668–1673, 1987; and in “Wavefront Reconstruction Using Iterative Fourier Transforms” as published in Applied Optics, Volume 30, No. 11, pages 1325–1327, 1991.
Recently an application of FFT-based algorithms for large adaptive optics systems was disclosed by Lisa Poyner et al. in “Fast Wavefront Reconstruction in Large Adaptive Optics Systems with use of the Fourier Transform”, as published in Journal of the Optical Society of America A, Volume 19, No. 10, pages 2100–2111, 2002.
For linear LS-based algorithms, Zou et al. proposed an efficient generalized algorithm with zero-padding of the slope data outside of the sampling pupil in “Generalized Wavefront Reconstruction Algorithm Applied in a Shack-Hartmann Test”, as published in Applied Optics, Volume 39, No. 2, pages 250–268, 2000. This algorithm is efficient given that it uses a regular and symmetrical reconstruction matrix as well as the efficient Cholesky Decomposition method in solving the large sparse linear equation set. However, the wavefront reconstructed with this algorithm leads to up to λ/4 deviation errors (peak-to-valley) with a corresponding rms error of about λ/14 from the original. While the deviation errors vary with the smoothness of the wavefront under construction. The λ/4 deviation errors may not be acceptable for many optical tests.
Recently Ellerbroek disclosed a minimum-variance wavefront reconstructor for adaptive optics utilizing sparse matrix techniques in “Efficient Computation of Minimum-Variance Wavefront Reconstructors with Sparse Matrix Techniques”, as published in the Journal of the Optical Society of America A, Volume 19, No. 9, pages 1803–1816, 2002. A multigrid preconditioned conjugate-gradient method was proposed by Gilles et al. for the reconstructor computation in “Multi-grid preconditioned conjugated-gradient method for large-scale wavefront reconstruction” as published in the Journal of the Optical Society of America A, Volume 19, No. 9, pages 1817–1822, 2002.
These methods are efficient for large adaptive optics (AO) systems and multi-conjugated adaptive optics (MCAO) systems and may also find application to wavefront estimation given that such step is implicit to the wavefront reconstructor. Also, MacMartin recently disclosed a local, hierarchic and iterative reconstructor for adaptive optics in “Local, Hierachic, and Iterative Reconstructors for Adaptive Optics,” as published in the Journal of the Optical Society of America, Volume 20, No 6, pages 1084–1093 (2003), which is related to the multigrid preconditioning method used in Gilles et al. MacMartin's algorithm, which is based on modal estimation, shows excellent (i.e. results approach relatively closely an optimal least-squares solution) relative performance across the Zernike basis function used to estimate the wavefront.
A main difference between wavefront estimation for AO and optical testing is that estimation errors above the bandwidth of the control loop can not be corrected in AO, and the common geometry used in AO is the Fried geometry. Nevertherless, several of the computationally efficient algorithms for wavefront reconstruction developed in the context of AO may find application to the optical testing problem with, in the case where modal estimation was used, an adjustment of the basis functions to satisfy orthogonality conditions across the pupil shape.
However, most of the algorithms in the prior art require setting up of the reconstruction procedure (e.g. setting up of matrices) before they can accept wavefront sensor (WFS) measurements. A process that could be fully automated is desirable in optical testing because of the frequent change in pupil sizes and shapes.
Also, the number of sampling grid points will vary significantly from a grid as small as a 4×4 points to thousands of points depending on the local curvature of the piece under test and its physical size. The trade-off is that a non-automatic procedure can perhaps better capitalize on the specific problem and optimize the reconstructor for speed for a given pupil shape and size. In cases where speed is not absolutely critical for a given pupil shape and size, but rather accuracy is the dominant performance metric and within a day various pupil shapes and sizes are tested, the establishment of a universal matrix that can accept any dataset from any pupil shape and size would be a tremendous gain. While a strength of the linear LS approach by Zou et al. is a universal normal matrix of wavefront reconstruction that provides immediate plug-in of WFS measurements for any pupil shapes and pupil sizes, the associated algorithm suffers remarkable deviation errors.
The present invention improves on the Zou et al. algorithm by reducing the deviation errors with Gerchberg-type iterations that enable extrapolating the slope data outside of the pupil to satisfy continuity boundary conditions. The approach of employing Gerchberg-type iterations in wavefront reconstruction, as Roddier et al. did with a DFTs-based method, is remarkable when combined with a linear LS method as disclosed in this invention in terms of the accuracy achieved and the convergence rate towards a solution. Moreover, because the linear LS approach yields a universal matrix regardless of the size of the input slope data set, it provides practical convenience in optical testing applications.